1. ## Sequence proof

If you have a sequence a_n which is bounded below and a sequence b_n which is bounded below, show that a_n + b_n is bounded below. The definition of bounded below is a_n > -M.

2. Originally Posted by natester
If you have a sequence a_n which is bounded below and a sequence b_n which is bounded below, show that a_n + b_n is bounded below. The definition of bounded below is a_n > -M.
your definition of being bounded below seems to be lacking. anyway, if that's what you are given, here is how we proceed. (i will write M instead of -M, 'cause that looks weird).

so we have $a_n > M$ and $b_n > K$ for all $n$.

thus, $a_n + b_n > M + K$ for all n. so $a_n + b_n$ is bounded below by M + K

3. That makes since to me, but how would I prove that the statement a_n + b_n > M + K is true?

4. Originally Posted by natester
That makes since to me, but how would I prove that the statement a_n + b_n > M + K is true?
um, it's not really something you have to prove. unless you want to go down to some kind of axiomatic level of number theory, which i doubt. it is just common sense that if you diminish each term, the result will be smaller

like since 2 > 1 and 3 > 1 it just makes sense that 2 + 3 > 1 + 1, because in place of two larger numbers, we put two smaller numbers, so the result must be smaller ...